Jacob Campbell, Department of Pure Mathematics, University of Waterloo
"Finitary approximations of free probability, involving combinatorial representation theory"
This
thesis
contributes
to
two
theories
which
approximate
free
probability
by
finitary
combinatorial
structures.
The
first
is
finite
free
probability,
which
is
concerned
with
expected
characteristic
polynomials
of
various
random
matrices
and
was
initiated
by
Marcus,
Spielman,
and
Srivastava
in
2015.
An
alternate
approach
to
some
of
their
results
for
sums
and
products
of
randomly
rotated
matrices
is
presented,
using
techniques
from
combinatorial
representation
theory.
Those
techniques
are
then
applied
to
the
commutators
of
such
matrices,
uncovering
the
non-trivial
but
tractable
combinatorics
of
immanants
and
Schur
polynomials.
The
second
is
the
connection
between
symmetric
groups
and
random
matrices,
specifically
the
asymptotics
of
star-transpositions
in
the
infinite
symmetric
group
and
the
gaussian
unitary
ensemble
(GUE).
For
a
continuous
family
of
factor
representations
of
$S_{\infty}$,
a
central
limit
theorem
for
the
star-transpositions
$(1,n)$
is
derived
from
the
insight
of
Gohm-K\"{o}stler
that
they
form
an
exchangeable
sequence
of
noncommutative
random
variables.
Then,
the
central
limit
law
is
described
by
a
random
matrix
model
which
continuously
deforms
the
well-known
traceless
GUE
by
taking
its
gaussian
entries
from
noncommutative
operator
algebras
with
canonical
commutation
relations
(CCR).
This
random
matrix
model
generalizes
results
of
K\"{o}stler
and
Nica
from
2021,
which
in
turn
generalized
a
result
of
Biane
from
1995.
Online (Please contact j48campb@uwaterloo.ca for details on how to attend.)